https://file.aiquickdraw.com/mathbot/file/1726623490632ekij6rdz.jpg

The graph provided in the image resembles the square root function, which is part of the toolkit functions. Specifically, the general form of a square root function is:

\[
y = a\sqrt{x - h} + k
\]

Where:
- \( a \) controls the vertical stretch or compression.
- \( h \) shifts the graph horizontally.
- \( k \) shifts the graph vertically.

### Identifying Transformations:
1. **Parent function**: The parent function is \( y = \sqrt{x} \), which starts at the origin (0,0).
2. **Shifts**: Based on the graph, the function appears to have been shifted **right** to \( h = -3 \) since the curve starts near \((-3, 0)\).
3. **Stretch/compression**: There doesn't seem to be much vertical stretch or compression, so \( a = 1 \).
4. **Vertical shift**: The graph does not appear to be shifted up or down significantly, so \( k = 0 \).

### The equation:
Thus, the equation for the graph would be:

\[
y = \sqrt{x + 3}
\]

This represents a square root function that has been shifted 3 units to the left.

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Would you like more details on how this transformation was identified or further explanation on specific parts of this process?

### Related Questions:
1. How does changing the value of \( a \) affect the graph of a square root function?
2. What are the differences between horizontal and vertical shifts in transformations?
3. How can I identify the direction of a shift when given a graph of a function?
4. Can vertical stretches and compressions occur simultaneously with shifts in both directions?
5. How would the equation change if the graph included a reflection?

### Tip:
Always compare the graph's key features, such as starting points and shape, to the parent function to determine the transformations applied.

Write an equation for the graphed function by using transformations of the graphs of one of the toolkit functions.

Learn how to write the equation for a square root function using graph transformations. The problem involves horizontal shifts and key features like starting points and shape comparisons to the parent function. Suitable for Grades 9-11.

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